Math 414: Analysis I
Homework 9
Due: April 14, 2014
Name:
The following problems are for additional practice and are not to be turned in: (All
problems come from Basic Analysis, Lebl )
Exercises: 3.2.1–3.2.3, 3.2.9, 3.2.12, 3.3.1, 3.3.2, 3.3.10
Turn in the following problems.
1. Exercise 3.2.4 in Basic Analysis, Lebl
2. Exercise 3.2.5 in Basic Analysis, Lebl
3. Exercise 3.2.10 in Basic Analysis, Lebl
4. Exercise 3.2.11 in Basic Analysis, Lebl
5. Exercise 3.3.1 in Basic Analysis, Lebl
6. Exercise 3.3.7 in Basic Analysis, Lebl
7. Define f : (0, ∞) → (0, ∞) by f (x) = (sin (1/x))2 . Prove that f is a continuous
function on (0, ∞).
Hint: Recall the the composition of continuous functions is continuous. Also to prove
that sin(x) is a continuous function on R, you may use that fact that|sin(x)| ≤
|x|
x+y
x−y
(Why?) and the product-to-sum identity: sin(x) − sin(y) = 2 sin 2 cos 2 .
√
8. Determine an interval [a, b] for which the function f (x) = 2 ln x + x − 2 has a root.
Prove that your claim is correct.
9. Bonus: Consider Thomae’s Function as given by
f (x) =
: x = pq , p, q ∈ Z, gcd(p, q) = 1, q > 0


1
q

0:
x∈
/Q
Prove that f is discontinuous at the rationals and continuous at the irrationals.
Hint: Suppose that b is an irrational number. Given an > 0, I can find an n0 ∈ N
such that 1/n0 < (Why?). Note that there are only a finite numbers of rationals
with denominator less than n0 on the interval (b − 1, b + 1) Why? Thus, you can
choose δ > 0 small enough so that (b − δ, b + δ) contains no rational numbers with
denominator less than n0 . Thus if |x − b| < δ, then |f (x) − f (b)| ≤ 1/n0 . (Why?)
Conclude that f is continuous at b.
1